The canonical-basis characterization of Calogero-Moser cellular characters

Let $G(l,1,n)$ be the imprimitive complex reflection group, let $c$ be a parameter satisfying the paper's standing hypothesis, and let $V_{\mathbf{s}}$ be the corresponding Fock-space representation. Let a vector of height $n$ mean a canonical basis vector lying in the degree-$n$ component. Canonical-basis cellular-character conjecture. The set of Calogero–Moser cellular characters of $G(l,1,n)$ for the parameter $c$ coincides with the set of characters obtained by evaluating at $q=1$ the vectors of height $n$ in the canonical basis of $V_{\mathbf{s}}$. The claim proposes a direct representation-theoretic interpretation of canonical basis vectors; the surrounding text presents it as a conjecture and relates quasimonomial specializations to sums of Calogero–Moser cellular characters.